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Abstract

Cyclotron resonance study of HgTe/CdTe-based quantum wells with both inverted and
normal band structures in quantizing magnetic fields was performed. In semimetallic
HgTe quantum wells with inverted band structure, a hole cyclotron resonance line was
observed for the first time. In the samples with normal band structure, interband
transitions were observed with wide line width due to quantum well width fluctuations.
In all samples, impurity-related magnetoabsorption lines were revealed. The obtained
results were interpreted within the Kane 8·8 model, the valence band offset of CdTe
and HgTe, and the Kane parameter EP being adjusted.

Keywords:

Background

HgTe/CdTe-based quantum wells (QWs) exhibit a number of remarkable properties. At
the critical HgTe QW thickness (6.3 to 7 nm depending on Cd content in the barrier),
the forbidden gap is absent and both electrons and holes are characterized by the
linear energy-momentum law of massless Dirac fermions [1,2]. When HgTe QW width exceeds this critical value, the energy band structure is inverted
(the conduction band states are formed by p-type wavefunctions while s-type wavefunctions
form the valence band states; see, e.g., [1,3] and references therein). In the inverted band structure regime, HgTe QWs are shown
to be two-dimensional (2D) topological insulators that have attracted a great fundamental
interest [1,2,4,5]. It was demonstrated [4] that a quantum spin Hall insulator state exists in such systems that can be destroyed
by magnetic field due to crossing of Landau levels of different bands [6]. Actually, these two levels have recently shown to display the effect of the avoided
crossing [7,8]. Hole-like symmetry of conduction-band Bloch functions enhances spin-dependent effects
like the Rashba splitting that has been shown to achieve 30 meV [3,9]. Wide HgTe/CdTe QWs have an indirect band structure [10]. If the well is wide (above 12.5 nm), the side maxima of the valence band overlap
with the conduction band. Then, the Fermi level can cross both valence and conduction
bands and a semimetallic state can be implemented which has been revealed by magnetotransport
measurements [11,12]. On the other hand, narrow HgTe QWs have been proposed as a material for detectors
of THz radiation since they possess certain advantages over bulk HgCdTe solid solutions
that are widely used for mid-infrared (IR) photodetectors. An alternative way to tune
the QW structure is to admix Cd into a wide HgTe QW. In [13,14], 30-nm-wide Hg1-xCdxTe QWs with a Cd content x > 0.13 are shown to have normal band structure. However, properties of such wells
are not identical to those of normal-band-structure HgTe QWs with the same bandgap,
namely wide HgCdTe QWs demonstrate indirect band structure, i.e., the side maximum
in the valence band exceeds that in the center of the Brillouin zone. An informative
method to probe the energy band structure both in bulk semiconductors and in QWs is
the cyclotron resonance (CR) technique. However, at the moment, there have been no
systematic studies on CR in HgTe/CdTe QWs with different band structures (cf.[6-9,13-19]). In this work, we present the first results on CR measurements in a semimetallic
sample with wide HgTe QW (inverted band structure) as well as in two samples with
normal band structures: narrow HgTe QW (for the first time) and wide HgCdTe (about
15% of cadmium).

Methods

Experimental

The structures under investigation were grown by molecular beam epitaxy on semi-insulating
GaAs(013) substrates [20]. The ZnTe and thick relaxed CdTe buffer layers, a 100-nm CdyHg1-yTe lower barrier layer, a Hg1-xCdxTe QW, and a 100-nm CdyHg1-yTe upper barrier layer were grown one by one, followed by a 50-nm CdTe cap layer (see
Table 1). The structures were not intentionally doped. In all samples, there were 2D carriers
in QWs at liquid helium temperature at dark conditions. Sample 1 was semimetallic,
which was confirmed by transport measurements. In sample 2, the dark electron concentration
was about 4×1010 cm-2 and it can be raised up to 1011 cm-2 by visible (or near-IR) light illumination (positive persistent photoconductivity
(PPC)). In sample 3, the dark electron concentration was about 1011 cm-2, but, in contrast to the previous sample, visible (or near-IR) light illumination
resulted to the concentration decrease down to complete freezing of free carriers
(negative PPC).

CR studies were carried out at T=4.2 K on 5×5 mm samples placed in the liquid helium. We used two superconducting
coils having maximum magnetic fields of 3 and 11 T. CR spectra were measured in the
Faraday configuration in two ways: by sweeping the magnetic field up to 3 T at a constant
frequency of the terahertz radiation and in a static magnetic field up to 11 T. In
the first case, the radiation was generated using quantum cascade lasers (QCLs) operating
at 2.6, 3.2, and 4.3 THz (pulse length 10 μs, repetition rate 5 to 10 kHz). The radiation transmitted through the sample was
detected using a Ge:Ga impurity photodetector. In the second case, a BRUKER 113V Fourier
transform (FT) spectrometer (Bruker Optik GmbH, Ettlingen, Germany) was used with
a globar radiation source. The spectral resolution was 4 cm-1. The transmitted radiation was detected by a composite bolometer. The measured spectra
presented here were normalized by sample transmission at B=0 and then divided by the rate of reference signals (signal without sample) at nonzero
and zero magnetic fields. The latter enables us to eliminate the influence of the
magnetic field on the bolometer sensitivity.

CR measurements in static magnetic fields up to 11 T were carried out in the Laboratoire
National des Champs Magnétiques Intenses in Grenoble (LNCMI-G). All other measurements
were performed at the Institute for Physics of Microstructures in Nizhny Novgorod.

Theoretical calculations

The band structure in the absence of the magnetic field and the Landau levels (LLs)
in the QWs under study were calculated in the axial approximation in the same way
as described in [19,21] in the four-band model. The calculation is based on the envelope function method
proposed by Burt [22]. The envelope functions were found as the solutions of the time-independent Schrödinger
equation with the 8·8 Hamiltonian taking into account a built-in strain. To calculate
the envelope functions and the corresponding values of the electron energy, the structure
was approximated by a superlattice of weakly interacting QWs. The lattice period was
chosen such that the interaction between the wells would not significantly affect
the energy spectrum. The calculation was performed by expanding the envelope functions
in plane waves. The expression for the Hamiltonian of the heterostructure grown on
the (013) plane was derived by the method described in [23]. The components of the built-in strain tensor were calculated with the use of the
formulas from [21]. The band parameters of the materials used in the calculation are taken from [21]. Two parameters of the model were adjusted to get better agreement between calculated
and measured transition energies. The first parameter is the valence band offset of
CdTe and HgTe. This parameter is not known well and, according to [23], is 570 ± 60 meV. We have used the value 620 meV. The second parameter is the Kane
parameter EPwhich is the same for both materials according to the model used. We assumed EP as 20.8 eV (instead of 18.8 eV [23]). The difference between the results of our calculations with ‘traditional’ and ‘adjusted’
parameters is shown on fan charts for all three samples under study. The dependences
of all parameters, except the bandgap, on the content of the solid solution Hg1-xCdxTe were assumed to be linear in x. The concentration dependence of the bandgap was described by the formula from [23]. It should be noted that the axial approximation we used is quite good for the conduction
band but can give a small error for the valence band (see, for example, Figure one
in [2]). According to our estimations, using axial approximation could result in the error
in LL energies in the valence band up to 2 meV (16 cm-1).

Figure 1 exemplifies the fan chart of calculated LLs in sample 1. According to calculation
results of the energy band spectra in the absence of the magnetic field in this sample,
there is a strong overlapping between the conduction band and the side maxima in the
valence band. As easy to see from Figure 1, in this sample, in addition to crossing of the lowest LL n = − 2 in the conduction band and the ‘top’ LL n = 0 in the valence band (typical for narrow QWs with inverted band structure [6,7,19]), LLs with high numbers in the valence band indeed overlap with those in the conduction
band. Therefore, at the Fermi level position between 40 and 80 cm-1, this structure would be semimetallic with 2D electrons and holes coexisting in HgTe
QW at the thermal equilibrium.

Results and discussion

Figure 2 presents typical CR spectra in sample 1 with inverted band structure obtained with
a FT spectrometer. The positions of all observed absorption peaks versus the magnetic
field are plotted in Figure 3. The symbol size characterizes the line intensity: bigger points correspond to more
intense lines. The calculated energies of allowed transitions between LLs (Δn=1) are also plotted in Figure 3. There are two stronger lines in the spectra: line β and line Π. In high magnetic fields, line β definitely arises from the transition between n = − 2 and n = − 1 LLs (cf.[6,7,19]). In this case, LL n = − 2 is fully occupied, level n = − 1 is empty (see Figure 1), and a transition is allowed, so we have a strong line in the spectra. In moderate magnetic
fields, line β can also be attributed to a transition in the conduction band: at B<4 T, energies of and transitions are closed to each other; as the magnetic field decreases, the occupancy
of LL n = − 1 in the conduction band in the semimetallic sample 1 increases, so the intensity
of the transition goes down while that of the one increases. Weak line βi, observed in high magnetic fields below line β, in our opinion, can be attributed to electron transitions between LL n = − 2 and residual donor states pertained to LL n = − 1.

Figure 2.Typical CR spectra for sample 1. The numbers against the CR lines are the magnetic field values in Tesla. Gray stripes
are Reststrahlen bands.

Figure 3.Energies of cyclotron transitions versus the magnetic field for sample 1. Solid lines correspond to the calculated transitions with adjusted parameters; thin
dotted lines, with traditional parameters. Symbols are experimental data. The size
of symbols indicates the intensity of CR lines: the smallest symbols correspond to
weak lines.

The second strong line Π is a hole CR apparently. It crosses X-axes in a nonzero magnetic field (≈5 T), which means that the transition takes place
between LLs crossing approximately in this field. The only allowed transition satisfying
this condition is the one in the valence band. Some discrepancy between measured and calculated energies
(see Figure 3) is due to violation of axial approximation. Thus, line Π is the first observed hole CR in HgTe QWs in quantizing magnetic fields. A weaker
line Πican be, by analogy, attributed to the transitions between the filled LL n = 1 in the valence band and impurity state pertained to empty LL n = 0.

In the magnetic field range 3.5 to 5 T in the CR spectra in sample 1, we have observed
a weaker line α that is known to result from the interband transition [6,7,19]. In B < 3.5 T, LL n = 1 seems to be occupied and the absorption decreases, while in B>5 T, the ‘initial’ level n = 0 seems to rise over the Fermi level.

Weak high-frequency lines I1 to I3 probably resulted from some interband transitions (cyclotron or impurity). At the
moment, it is difficult to identify them only because of the great number of allowed
transitions between valence and conduction band LLs in this frequency range. At last,
the line U whose spectral position does not depend on the magnetic field most probably
resulted from transitions between impurity states pertained to LLs n = − 2 in the valence and conduction bands (since direct transitions between these
two LLs are forbidden in the Faraday configuration).

Investigations of CR absorptions in sample 2 also revealed a lot of spectral features.
In this sample, in addition to the magnetoabsorption study with a FT spectrometer,
we also measured CR with QCLs at different 2D electron concentrations varied using
the positive PPC effect. As easy to see from Figure 4, the rise of the electron concentration results in the increase of the CR line intensity
only while its position is unchanged. This is an indication that the observed CR line
resulted from transitions from one and the same LL (namely n = 1 in the conduction band; see Figure 5) because in classical magnetic fields, a gradual shift of the CR line to higher magnetic
fields with the concentration increase is observed [19].

Figure 4.Typical CR spectra for sample 2 measured using 3.2-THz QCL. In the absence of visible light illumination (1) and at various levels of illumination
(2 to 5). The carrier density in units of 1010 cm-2 is 3.5 (1), 5.4 (2), 7.2 (3), 9.3 (4), and 10.3 (5).

Figure 5.Energies of cyclotron transitions versus the magnetic field for sample 2. Solid lines correspond to the calculated transitions with adjusted parameters; thin
dotted lines, with traditional parameters. Symbols are experimental data. The size
of symbols indicates the intensity of CR lines: the smallest symbols correspond to
weak lines.

Experimental data obtained in sample 2 with both the FT spectrometer and the QCLs,
as well as calculated energies of allowed transitions between conduction band LLs
versus magnetic field are presented in Figure 5. It is clearly seen that the data obtained with different techniques correspond fairly
well (see lower left corner in Figure 5). Besides, using QCL operating at 4.3 THz made it possible to measure CR in the phonon
absorption band around 150 cm-1 (see Figure 2) due to a high stability of QCL radiation intensity.

The main lines in absorption spectra in sample 2 are α, γ, and δ. This sample has a normal band structure; therefore, all the transitions take place
within the conduction band. The LL structure is analogous to that of sample 100708
studied earlier (see Figure one in [14]). Line α corresponds to the transition from the lowest LL in the conduction band. In high magnetic fields over 4 T, the
LL filling factor is less than unity and all the electrons in the QW occupy LL n=0; therefore, only CR line α is observed. However, in lower magnetic fields, the electrons populate the next LL
n = − 1 (see Figure one in [14]) and the transitions (line γ) are observed. At still smaller magnetic fields, the third LL in the conduction band
is occupied that leads to a decrease in the intensity of transition (line α) and in the appearance of line δ(transition ).

The observed intensive absorption line γ− is to be considered separately. Its position corresponds fairly well to the transition
between two lowest LLs . In magnetic fields over 5.5 T, where this line is observed, LL n = 0 is filled while that of n = − 1 is empty. However, according to our calculations within the axial model, the
square of the electrodipole matrix element for this transition is by 4 orders of magnitude
less than that for transition (line α). Actually, the transition corresponds to electron spin resonance that should not be observed in
the Faraday configuration. Nevertheless, line γ− is clearly seen in the absorption spectra. Probably, this line resulted from transitions
between shallow-donor impurity states pertained to LLs 0 and − 1. It is also possible
that because of the absence of the axial symmetry in reality, the square of the matrix
element for this transition will be significantly higher. In any case, the origin
of line γ− (which has been observed in a number of samples with normal band structure) requires
further investigations.

Weak line αi seems to result from the transition between the 1s-like state of residual shallow donors pertained to LL n = 0 and the excited 2p + -like state pertained to LL n = 1. In contrast to impurity lines βi and Πi observed in sample 1 with inverted band structure (Figure 3), the energy of the transitions corresponding to line αi exceeds that of line α since the binding energy of the 1s-like ground state is greater than that of the excited 2p+-like state. The origin of other weak lines observed in the absorption spectra in
sample 2 requires further studies.

The last sample 3 under study contains narrow HgTe QW with nominal width of 6.3 nm
that should correspond to zero bandgap [1,2]. In contrast to the previous one, this sample demonstrated negative persistent photoconductivity
at illumination by visible light down to electron freezing out. The latter enables
us to measure the spectrum of interband photoconductivity (Figure 6; cf.[13]). One can see a distinct low-frequency edge of the conductivity at 380 cm-1 (47 meV). According to the theoretical model used, the gap value of 47 meV corresponds
to a significantly narrower QW with normal band structure. Therefore, general features
of the LL fan chart in this sample are the same as those in sample 2 (see Figure one
in [14]).

Figure 6.Typical CR spectra and photoconductivity spectrum for sample 3. The numbers against the CR lines are the magnetic field values in Tesla. Arrows indicate
the observed cyclotron peaks. Gray stripes are Reststrahlen bands. The ‘bandgap’ mark
indicates the value of the bandgap for 4.8-nm HgTe QW.

Typical CR spectra in sample 3 are plotted in Figure 6, and the overall data are presented in Figure 7 together with calculated energies of CR transitions versus the magnetic field. There
are four main lines in the spectra to be considered: α, β, U1, and U2. The nature of lines α and β is well known. As discussed above, line αcorresponds to the transition from the lowest LL in the conduction band . Line β corresponds to the interband transition from the top LL in the valence band to the
conduction band . The intensity of this line decreases in magnetic fields below 3 T (see Figure 7) because of partial filling of the ‘final’ LL for this transition n = − 1. To the best of our knowledge, this is the first observation on the interband
transition in the HgTe QW with normal band structure. At present, such transitions
have been observed in HgTe QWs with inverted band structure only (see, e.g., [6,7,19]). It should be mentioned that the extrapolation of the spectral position of line
β to B = 0 gives slightly less bandgap (340 cm-1) than that obtained from the photoconductivity spectrum (measured on another sample
cut from the same wafer). Therefore, in our calculations, we used a compromised (between
CR and photoconductivity data) QW width of 4.8 nm. Let us note that in this sample
3 with the narrowest QW, the linewidth of the interband transition (β) exceeds significantly that of the intraband one (α), while in broad QWs, they are approximately the same (see, e.g., Figure 2; [7,19]). To our opinion, a spreading of the interband line β in sample 3 with narrow QW resulted from the enhanced role of one-monolayer fluctuations
(about 0.5 nm) of this narrow QW width which, in turn, leads to bandgap fluctuations.

Figure 7.Energies of cyclotron transitions versus the magnetic field for sample 3. Solid lines correspond to the calculated transitions with adjusted parameters; thin
dotted lines, with traditional parameters. Symbols are experimental data. The size
of symbols indicates the intensity of CR lines: the smallest symbols correspond to
weak lines.

The nature of the most intense low-frequency line in CR spectra U1 is not quite clear. It persists up to the maximal magnetic field used (11 T) when
the LL filling factor is much less than unity, so it cannot be attributed to transitions
between higher LLs in the conduction band. On the other hand, the transition energies
are much less than the bandgap. Therefore, the only reasonable explanation is to attribute
this absorption line to intracenter excitation of residual donors. In wide QWs (such
as in sample 2), the shallow-donor binding energies are small compared to those of
the CR ones because of small electron effective masses (of the order 10-2m0, where m0 is the free electron mass). However, in narrow QWs, the donor binding energies increase
significantly since the QW potential pushes the donor wavefunction to the impurity
ion. A weaker absorption line U2 seems to result from some impurity interband transition since the line is as broad
as line β. As a whole, the accordance between measured and calculated data in sample 3 with
the narrowest QW (Figure 7) is worst than those in samples 1 and 2 with wider QWs. The latter means that the
theoretical model for the description of such narrow QWs is to be elaborated.

Conclusions

In conclusion, we have measured CR in a set of nominally undoped HgCdTe QWs with different
band structures in quantizing magnetic fields. The results obtained are interpreted
on the basis of Landau level calculations within the Kane 8·8 model. In wide semimetallic
HgTe QWs with inverted band structure, both intra- and interband transitions between
Landau levels are identified, the CR line being accompanied by impurity satellites.
A hole CR line has been observed for the first time. In two samples with normal band
structure: wide (30 nm) HgCdTe QW and narrow (4.8 nm) HgTe QW, interband CR transitions
have been revealed in the spectra, the interband absorption line width in the narrow
QW being spread due to QW width fluctuations. The adjusted material parameters: valence
band offset of CdTe and HgTe 620 meV (instead of 570 meV) and the Kane parameter EP20.8 eV (instead of 18.8 eV), are proposed from the comparison of the experimental
and calculation data.

Abbreviations

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MSZ, FT, and MO performed the CR measurements using the Fourier transform spectrometer.
MSZ carried out all calculations. AVI and KVM performed the CR measurements with QCL.
AVI performed the photoconductivity measurements. KES characterized the samples with
magnetotransport measurements. SAD and NNM grew the experimental samples with MBE
technology. AVI, VIG, WK, and MSZ explained the obtained data. AVI and VIG wrote the
manuscript draft. All authors read and approved the final manuscript.

Acknowledgements

We are grateful to Yu.G. Sadof’ev and Trion Technology Inc., USA, for providing us
with quantum cascade lasers. This work was supported by the Russian Foundation for
Basic Research (grants 11-02-00958, 11-02-97061, and 11-02-93111), the Ministry of
Education and Science of the Russian Federation (state contract nos. 16.740.11.0321
and 16.518.11.7018), the Council of the President of the Russian Federation for Support
of Young Scientists and Leading Scientific Schools (project nos. MK-1114.2011.2 and
NSh-4756.2012.2), and the Russian Academy of Sciences. The Montpellier team would
also like to acknowledge the CNRS via GDR-I project ‘Semiconductor sources and detectors
of THz frequencies’ and the GIS-Teralab.